# Intuition of law of total variance.

General question: The law of total variance is given by $$Var(Y)=E[Var(Y|X)]+Var(E[Y|X])$$

Question: I know variance is how spread out variables are, but the formula doesn't make sense intuitively, can some one explain from the ground up? Thanks.

• In what sense do you find it making no sense intuitively? It says to me that the variance of $Y$ is a combination of the expected variance conditioned on $X$ and the variance caused by changes in $X$, which is at least plausible. It also seems consistent with saying $Var(aY+b)=a^2\, Var(Y)$ Mar 1 '20 at 23:36
• For me, it's best understood by considering the case $Y = Z + X$ for independent $Z,X$. Then $$E[\text{Var}(Y|X)] = E[\text{Var}(Z)] = \text{Var}(Z) \\ \text{Var}(E[Y|X]) = \text{Var}(E[Z] + X) = \text{Var}(X)$$ which matches with the known $\text{Var}(Z+X) = \text{Var}(Z) + \text{Var}(X)$ Mar 1 '20 at 23:36
• Thanks for the explanation, and I found this link helpful to mine own question: math.stackexchange.com/questions/1742578/… Mar 1 '20 at 23:46